Tool CaT
stdout:
MAYBE
Problem:
minus(x,0()) -> x
minus(0(),y) -> 0()
minus(s(x),s(y)) -> minus(p(s(x)),p(s(y)))
minus(x,plus(y,z)) -> minus(minus(x,y),z)
p(s(s(x))) -> s(p(s(x)))
p(0()) -> s(s(0()))
div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
div(plus(x,y),z) -> plus(div(x,z),div(y,z))
plus(0(),y) -> y
plus(s(x),y) -> s(plus(y,minus(s(x),s(0()))))
Proof:
OpenTool IRC1
stdout:
MAYBE
Tool IRC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ minus(x, 0()) -> x
, minus(0(), y) -> 0()
, minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
, minus(x, plus(y, z)) -> minus(minus(x, y), z)
, p(s(s(x))) -> s(p(s(x)))
, p(0()) -> s(s(0()))
, div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
, div(plus(x, y), z) -> plus(div(x, z), div(y, z))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(y, minus(s(x), s(0()))))}
Proof Output:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: minus^#(x, 0()) -> c_0()
, 2: minus^#(0(), y) -> c_1()
, 3: minus^#(s(x), s(y)) -> c_2(minus^#(p(s(x)), p(s(y))))
, 4: minus^#(x, plus(y, z)) -> c_3(minus^#(minus(x, y), z))
, 5: p^#(s(s(x))) -> c_4(p^#(s(x)))
, 6: p^#(0()) -> c_5()
, 7: div^#(s(x), s(y)) -> c_6(div^#(minus(x, y), s(y)))
, 8: div^#(plus(x, y), z) -> c_7(plus^#(div(x, z), div(y, z)))
, 9: plus^#(0(), y) -> c_8()
, 10: plus^#(s(x), y) -> c_9(plus^#(y, minus(s(x), s(0()))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{7} [ inherited ]
|
`->{8} [ inherited ]
|
|->{9} [ NA ]
|
`->{10} [ inherited ]
|
`->{9} [ NA ]
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(n^1)) ]
->{3,4} [ inherited ]
|
|->{1} [ NA ]
|
`->{2} [ MAYBE ]
Sub-problems:
-------------
* Path {3,4}: inherited
---------------------
This path is subsumed by the proof of path {3,4}->{1}.
* Path {3,4}->{1}: NA
-------------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(0(), y) -> 0()
, minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
, minus(x, plus(y, z)) -> minus(minus(x, y), z)
, p(s(s(x))) -> s(p(s(x)))
, p(0()) -> s(s(0()))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {3,4}->{2}: MAYBE
----------------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(0(), y) -> 0()
, minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
, minus(x, plus(y, z)) -> minus(minus(x, y), z)
, p(s(s(x))) -> s(p(s(x)))
, p(0()) -> s(s(0()))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ minus^#(s(x), s(y)) -> c_2(minus^#(p(s(x)), p(s(y))))
, minus^#(x, plus(y, z)) -> c_3(minus^#(minus(x, y), z))
, minus^#(0(), y) -> c_1()
, minus(x, 0()) -> x
, minus(0(), y) -> 0()
, minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
, minus(x, plus(y, z)) -> minus(minus(x, y), z)
, p(s(s(x))) -> s(p(s(x)))
, p(0()) -> s(s(0()))}
Proof Output:
The input cannot be shown compatible
* Path {5}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(p) = {}, Uargs(plus) = {},
Uargs(div) = {}, Uargs(minus^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {}, Uargs(p^#) = {}, Uargs(c_4) = {1},
Uargs(div^#) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(plus^#) = {}, Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
div(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[3 3] [0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
c_5() = [0]
[0]
div^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8() = [0]
[0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {p^#(s(s(x))) -> c_4(p^#(s(x)))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [1]
[0 0] [0]
p^#(x1) = [1 0] x1 + [0]
[0 0] [0]
c_4(x1) = [1 0] x1 + [0]
[0 0] [0]
* Path {6}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(p) = {}, Uargs(plus) = {},
Uargs(div) = {}, Uargs(minus^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {}, Uargs(p^#) = {}, Uargs(c_4) = {},
Uargs(div^#) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(plus^#) = {}, Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
div(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0() = [0]
[0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
div^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8() = [0]
[0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {p^#(0()) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(p^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
p^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_5() = [0]
[1]
* Path {7}: inherited
-------------------
This path is subsumed by the proof of path {7}->{8}->{10}->{9}.
* Path {7}->{8}: inherited
------------------------
This path is subsumed by the proof of path {7}->{8}->{10}->{9}.
* Path {7}->{8}->{9}: NA
----------------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(0(), y) -> 0()
, minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
, minus(x, plus(y, z)) -> minus(minus(x, y), z)
, p(s(s(x))) -> s(p(s(x)))
, p(0()) -> s(s(0()))
, div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
, div(plus(x, y), z) -> plus(div(x, z), div(y, z))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(y, minus(s(x), s(0()))))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {7}->{8}->{10}: inherited
------------------------------
This path is subsumed by the proof of path {7}->{8}->{10}->{9}.
* Path {7}->{8}->{10}->{9}: NA
----------------------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(0(), y) -> 0()
, minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
, minus(x, plus(y, z)) -> minus(minus(x, y), z)
, p(s(s(x))) -> s(p(s(x)))
, p(0()) -> s(s(0()))
, div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
, div(plus(x, y), z) -> plus(div(x, z), div(y, z))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(y, minus(s(x), s(0()))))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: minus^#(x, 0()) -> c_0()
, 2: minus^#(0(), y) -> c_1()
, 3: minus^#(s(x), s(y)) -> c_2(minus^#(p(s(x)), p(s(y))))
, 4: minus^#(x, plus(y, z)) -> c_3(minus^#(minus(x, y), z))
, 5: p^#(s(s(x))) -> c_4(p^#(s(x)))
, 6: p^#(0()) -> c_5()
, 7: div^#(s(x), s(y)) -> c_6(div^#(minus(x, y), s(y)))
, 8: div^#(plus(x, y), z) -> c_7(plus^#(div(x, z), div(y, z)))
, 9: plus^#(0(), y) -> c_8()
, 10: plus^#(s(x), y) -> c_9(plus^#(y, minus(s(x), s(0()))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{7} [ inherited ]
|
`->{8} [ inherited ]
|
|->{9} [ NA ]
|
`->{10} [ inherited ]
|
`->{9} [ NA ]
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(n^1)) ]
->{3,4} [ inherited ]
|
|->{1} [ NA ]
|
`->{2} [ MAYBE ]
Sub-problems:
-------------
* Path {3,4}: inherited
---------------------
This path is subsumed by the proof of path {3,4}->{1}.
* Path {3,4}->{1}: NA
-------------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(0(), y) -> 0()
, minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
, minus(x, plus(y, z)) -> minus(minus(x, y), z)
, p(s(s(x))) -> s(p(s(x)))
, p(0()) -> s(s(0()))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {3,4}->{2}: MAYBE
----------------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(0(), y) -> 0()
, minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
, minus(x, plus(y, z)) -> minus(minus(x, y), z)
, p(s(s(x))) -> s(p(s(x)))
, p(0()) -> s(s(0()))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ minus^#(s(x), s(y)) -> c_2(minus^#(p(s(x)), p(s(y))))
, minus^#(x, plus(y, z)) -> c_3(minus^#(minus(x, y), z))
, minus^#(0(), y) -> c_1()
, minus(x, 0()) -> x
, minus(0(), y) -> 0()
, minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
, minus(x, plus(y, z)) -> minus(minus(x, y), z)
, p(s(s(x))) -> s(p(s(x)))
, p(0()) -> s(s(0()))}
Proof Output:
The input cannot be shown compatible
* Path {5}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(p) = {}, Uargs(plus) = {},
Uargs(div) = {}, Uargs(minus^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {}, Uargs(p^#) = {}, Uargs(c_4) = {1},
Uargs(div^#) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(plus^#) = {}, Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
p(x1) = [0] x1 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
div(x1, x2) = [0] x1 + [0] x2 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
p^#(x1) = [3] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
div^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {p^#(s(s(x))) -> c_4(p^#(s(x)))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [4]
p^#(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [3]
* Path {6}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(p) = {}, Uargs(plus) = {},
Uargs(div) = {}, Uargs(minus^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {}, Uargs(p^#) = {}, Uargs(c_4) = {},
Uargs(div^#) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(plus^#) = {}, Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
p(x1) = [0] x1 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
div(x1, x2) = [0] x1 + [0] x2 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
div^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8() = [0]
c_9(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {p^#(0()) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(p^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
p^#(x1) = [1] x1 + [7]
c_5() = [1]
* Path {7}: inherited
-------------------
This path is subsumed by the proof of path {7}->{8}->{10}->{9}.
* Path {7}->{8}: inherited
------------------------
This path is subsumed by the proof of path {7}->{8}->{10}->{9}.
* Path {7}->{8}->{9}: NA
----------------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(0(), y) -> 0()
, minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
, minus(x, plus(y, z)) -> minus(minus(x, y), z)
, p(s(s(x))) -> s(p(s(x)))
, p(0()) -> s(s(0()))
, div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
, div(plus(x, y), z) -> plus(div(x, z), div(y, z))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(y, minus(s(x), s(0()))))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {7}->{8}->{10}: inherited
------------------------------
This path is subsumed by the proof of path {7}->{8}->{10}->{9}.
* Path {7}->{8}->{10}->{9}: NA
----------------------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(0(), y) -> 0()
, minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
, minus(x, plus(y, z)) -> minus(minus(x, y), z)
, p(s(s(x))) -> s(p(s(x)))
, p(0()) -> s(s(0()))
, div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
, div(plus(x, y), z) -> plus(div(x, z), div(y, z))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(y, minus(s(x), s(0()))))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
3) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
4) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
5) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
Tool RC1
stdout:
MAYBE
Tool RC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ minus(x, 0()) -> x
, minus(0(), y) -> 0()
, minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
, minus(x, plus(y, z)) -> minus(minus(x, y), z)
, p(s(s(x))) -> s(p(s(x)))
, p(0()) -> s(s(0()))
, div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
, div(plus(x, y), z) -> plus(div(x, z), div(y, z))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(y, minus(s(x), s(0()))))}
Proof Output:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: minus^#(x, 0()) -> c_0(x)
, 2: minus^#(0(), y) -> c_1()
, 3: minus^#(s(x), s(y)) -> c_2(minus^#(p(s(x)), p(s(y))))
, 4: minus^#(x, plus(y, z)) -> c_3(minus^#(minus(x, y), z))
, 5: p^#(s(s(x))) -> c_4(p^#(s(x)))
, 6: p^#(0()) -> c_5()
, 7: div^#(s(x), s(y)) -> c_6(div^#(minus(x, y), s(y)))
, 8: div^#(plus(x, y), z) -> c_7(plus^#(div(x, z), div(y, z)))
, 9: plus^#(0(), y) -> c_8(y)
, 10: plus^#(s(x), y) -> c_9(plus^#(y, minus(s(x), s(0()))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{7} [ inherited ]
|
`->{8} [ inherited ]
|
|->{9} [ NA ]
|
`->{10} [ inherited ]
|
`->{9} [ NA ]
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(n^1)) ]
->{3,4} [ inherited ]
|
|->{1} [ MAYBE ]
|
`->{2} [ NA ]
Sub-problems:
-------------
* Path {3,4}: inherited
---------------------
This path is subsumed by the proof of path {3,4}->{1}.
* Path {3,4}->{1}: MAYBE
----------------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(0(), y) -> 0()
, minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
, minus(x, plus(y, z)) -> minus(minus(x, y), z)
, p(s(s(x))) -> s(p(s(x)))
, p(0()) -> s(s(0()))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ minus^#(s(x), s(y)) -> c_2(minus^#(p(s(x)), p(s(y))))
, minus^#(x, plus(y, z)) -> c_3(minus^#(minus(x, y), z))
, minus^#(x, 0()) -> c_0(x)
, minus(x, 0()) -> x
, minus(0(), y) -> 0()
, minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
, minus(x, plus(y, z)) -> minus(minus(x, y), z)
, p(s(s(x))) -> s(p(s(x)))
, p(0()) -> s(s(0()))}
Proof Output:
The input cannot be shown compatible
* Path {3,4}->{2}: NA
-------------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(0(), y) -> 0()
, minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
, minus(x, plus(y, z)) -> minus(minus(x, y), z)
, p(s(s(x))) -> s(p(s(x)))
, p(0()) -> s(s(0()))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {5}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(p) = {}, Uargs(plus) = {},
Uargs(div) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(p^#) = {},
Uargs(c_4) = {1}, Uargs(div^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(plus^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
div(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[3 3] [0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [0]
c_5() = [0]
[0]
div^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {p^#(s(s(x))) -> c_4(p^#(s(x)))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [1]
[0 0] [0]
p^#(x1) = [1 0] x1 + [0]
[0 0] [0]
c_4(x1) = [1 0] x1 + [0]
[0 0] [0]
* Path {6}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(p) = {}, Uargs(plus) = {},
Uargs(div) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(p^#) = {}, Uargs(c_4) = {},
Uargs(div^#) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(plus^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
p(x1) = [0 0] x1 + [0]
[0 0] [0]
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
div(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
p^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4(x1) = [0 0] x1 + [0]
[0 0] [0]
c_5() = [0]
[0]
div^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_6(x1) = [0 0] x1 + [0]
[0 0] [0]
c_7(x1) = [0 0] x1 + [0]
[0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_8(x1) = [0 0] x1 + [0]
[0 0] [0]
c_9(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {p^#(0()) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(p^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [2]
[2]
p^#(x1) = [2 0] x1 + [7]
[2 2] [7]
c_5() = [0]
[1]
* Path {7}: inherited
-------------------
This path is subsumed by the proof of path {7}->{8}->{10}->{9}.
* Path {7}->{8}: inherited
------------------------
This path is subsumed by the proof of path {7}->{8}->{10}->{9}.
* Path {7}->{8}->{9}: NA
----------------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(0(), y) -> 0()
, minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
, minus(x, plus(y, z)) -> minus(minus(x, y), z)
, p(s(s(x))) -> s(p(s(x)))
, p(0()) -> s(s(0()))
, div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
, div(plus(x, y), z) -> plus(div(x, z), div(y, z))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(y, minus(s(x), s(0()))))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {7}->{8}->{10}: inherited
------------------------------
This path is subsumed by the proof of path {7}->{8}->{10}->{9}.
* Path {7}->{8}->{10}->{9}: NA
----------------------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(0(), y) -> 0()
, minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
, minus(x, plus(y, z)) -> minus(minus(x, y), z)
, p(s(s(x))) -> s(p(s(x)))
, p(0()) -> s(s(0()))
, div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
, div(plus(x, y), z) -> plus(div(x, z), div(y, z))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(y, minus(s(x), s(0()))))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: minus^#(x, 0()) -> c_0(x)
, 2: minus^#(0(), y) -> c_1()
, 3: minus^#(s(x), s(y)) -> c_2(minus^#(p(s(x)), p(s(y))))
, 4: minus^#(x, plus(y, z)) -> c_3(minus^#(minus(x, y), z))
, 5: p^#(s(s(x))) -> c_4(p^#(s(x)))
, 6: p^#(0()) -> c_5()
, 7: div^#(s(x), s(y)) -> c_6(div^#(minus(x, y), s(y)))
, 8: div^#(plus(x, y), z) -> c_7(plus^#(div(x, z), div(y, z)))
, 9: plus^#(0(), y) -> c_8(y)
, 10: plus^#(s(x), y) -> c_9(plus^#(y, minus(s(x), s(0()))))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{7} [ inherited ]
|
`->{8} [ inherited ]
|
|->{9} [ NA ]
|
`->{10} [ inherited ]
|
`->{9} [ NA ]
->{6} [ YES(?,O(1)) ]
->{5} [ YES(?,O(n^1)) ]
->{3,4} [ inherited ]
|
|->{1} [ MAYBE ]
|
`->{2} [ NA ]
Sub-problems:
-------------
* Path {3,4}: inherited
---------------------
This path is subsumed by the proof of path {3,4}->{1}.
* Path {3,4}->{1}: MAYBE
----------------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(0(), y) -> 0()
, minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
, minus(x, plus(y, z)) -> minus(minus(x, y), z)
, p(s(s(x))) -> s(p(s(x)))
, p(0()) -> s(s(0()))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ minus^#(s(x), s(y)) -> c_2(minus^#(p(s(x)), p(s(y))))
, minus^#(x, plus(y, z)) -> c_3(minus^#(minus(x, y), z))
, minus^#(x, 0()) -> c_0(x)
, minus(x, 0()) -> x
, minus(0(), y) -> 0()
, minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
, minus(x, plus(y, z)) -> minus(minus(x, y), z)
, p(s(s(x))) -> s(p(s(x)))
, p(0()) -> s(s(0()))}
Proof Output:
The input cannot be shown compatible
* Path {3,4}->{2}: NA
-------------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(0(), y) -> 0()
, minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
, minus(x, plus(y, z)) -> minus(minus(x, y), z)
, p(s(s(x))) -> s(p(s(x)))
, p(0()) -> s(s(0()))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {5}: YES(?,O(n^1))
-----------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(p) = {}, Uargs(plus) = {},
Uargs(div) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(p^#) = {},
Uargs(c_4) = {1}, Uargs(div^#) = {}, Uargs(c_6) = {},
Uargs(c_7) = {}, Uargs(plus^#) = {}, Uargs(c_8) = {},
Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
p(x1) = [0] x1 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
div(x1, x2) = [0] x1 + [0] x2 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
p^#(x1) = [3] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
div^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {p^#(s(s(x))) -> c_4(p^#(s(x)))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [4]
p^#(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [3]
* Path {6}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(minus) = {}, Uargs(s) = {}, Uargs(p) = {}, Uargs(plus) = {},
Uargs(div) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(p^#) = {}, Uargs(c_4) = {},
Uargs(div^#) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
Uargs(plus^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
minus(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
p(x1) = [0] x1 + [0]
plus(x1, x2) = [0] x1 + [0] x2 + [0]
div(x1, x2) = [0] x1 + [0] x2 + [0]
minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
div^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_8(x1) = [0] x1 + [0]
c_9(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {p^#(0()) -> c_5()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(p^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
0() = [7]
p^#(x1) = [1] x1 + [7]
c_5() = [1]
* Path {7}: inherited
-------------------
This path is subsumed by the proof of path {7}->{8}->{10}->{9}.
* Path {7}->{8}: inherited
------------------------
This path is subsumed by the proof of path {7}->{8}->{10}->{9}.
* Path {7}->{8}->{9}: NA
----------------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(0(), y) -> 0()
, minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
, minus(x, plus(y, z)) -> minus(minus(x, y), z)
, p(s(s(x))) -> s(p(s(x)))
, p(0()) -> s(s(0()))
, div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
, div(plus(x, y), z) -> plus(div(x, z), div(y, z))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(y, minus(s(x), s(0()))))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {7}->{8}->{10}: inherited
------------------------------
This path is subsumed by the proof of path {7}->{8}->{10}->{9}.
* Path {7}->{8}->{10}->{9}: NA
----------------------------
The usable rules for this path are:
{ minus(x, 0()) -> x
, minus(0(), y) -> 0()
, minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
, minus(x, plus(y, z)) -> minus(minus(x, y), z)
, p(s(s(x))) -> s(p(s(x)))
, p(0()) -> s(s(0()))
, div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
, div(plus(x, y), z) -> plus(div(x, z), div(y, z))
, plus(0(), y) -> y
, plus(s(x), y) -> s(plus(y, minus(s(x), s(0()))))}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
3) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
4) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
5) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.